Elsevier

NeuroImage

Volume 38, Issue 1, 15 October 2007, Pages 95-113
NeuroImage

A fast diffeomorphic image registration algorithm

https://doi.org/10.1016/j.neuroimage.2007.07.007Get rights and content

Abstract

This paper describes DARTEL, which is an algorithm for diffeomorphic image registration. It is implemented for both 2D and 3D image registration and has been formulated to include an option for estimating inverse consistent deformations. Nonlinear registration is considered as a local optimisation problem, which is solved using a Levenberg–Marquardt strategy. The necessary matrix solutions are obtained in reasonable time using a multigrid method. A constant Eulerian velocity framework is used, which allows a rapid scaling and squaring method to be used in the computations. DARTEL has been applied to intersubject registration of 471 whole brain images, and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects.

Introduction

At its simplest, image registration involves estimating a smooth, continuous mapping between the points in one image and those in another. The relative shapes of the images can then be determined from the parameters that encode the mapping. The objective is usually to determine the single “best” set of values for these parameters. There are many ways of modelling such mappings, but these fit into two broad categories of parameterisation (Miller et al., 1997).

  • The small-deformation framework does not necessarily preserve topology—although if the deformations are relatively small, then it may still be preserved.

  • The large-deformation framework generates deformations (diffeomorphisms) that have a number of elegant mathematical properties, such as enforcing the preservation of topology.

Many registration approaches still use a small deformation model. These models parameterise a displacement field (u), which is simply added to an identity transform (x).ϕ(x)=x+u(x)In such parameterisations, the inverse transformation is sometimes approximated by subtracting the displacement. It is worth noting that this is only a very approximate inverse, which fails badly for larger deformations. As shown in Fig. 1, compositions of these forward and “inverse” deformations do not produce an identity transform. Small deformation models do not necessarily enforce a one-to-one mapping, particularly if the model assumes the displacements are drawn from a multivariate Gaussian probability density.

The large-deformation or diffeomorphic setting is a much more elegant framework. A diffeomorphism is a globally one-to-one (objective) smooth and continuous mapping with derivatives that are invertible (i.e. nonzero Jacobian determinant). If the mapping is not diffeomorphic, then topology1 is not necessarily preserved. A key element of a diffeomorphic setting is that it enforces consistency under compositions of the deformations. A composition of two functions is essentially taking one function of the other in order to produce a new function. For two functions, Φ2 and Φ1 this would be denoted byϕ2ϕ1x=ϕ2(ϕ1(x))For deformations, the composition operation is achieved by resampling one deformation field by another.2 If the deformations are diffeomorphic, then the result of the composition will also be diffeomorphic. In reality though, deformations are generally represented discretely with a finite number of parameters, so there may be some small violations—particularly if the composition is done using low degree interpolation methods. Perfect (i.e. infinitely dimensional) diffeomorphisms form a Lie group under the composition operation, as they satisfy the requirements of closure, associativity, inverse and identity (see Fig. 2).

The early diffeomorphic registration approaches were based on the greedy “viscous fluid” registration method of Christensen et al., 1994, Christensen et al., 1996. In these models, finite difference methods are used to solve the differential equations that model one image as it “flows” to match the shape of the other. At the time, the advantage of these methods was that they were able to account for large displacements while ensuring that the topology of the warped image was preserved. They also provided a useful foundation from which later methods arose. Viscous fluid methods require the solutions to large sets of partial differential equations. The earliest implementations were computationally expensive because solving the equations used successive over-relaxation. Such relaxation methods are inefficient when there are large low frequency components to estimate. Since then, a number of faster ways of solving the differential equations have been devised. These include the use of Fourier transforms to convolve with the impulse response of the linear regularisation operator (Bro-Nielsen and Gramkow, 1996), or by convolving with a separable approximation (Thirion, 1995).

More recent algorithms for large deformation registration aim to find the smoothest possible solution. For example, the LDDMM (large deformation diffeomorphic metric mapping) algorithm (Beg et al., 2005) does not fix the deformation parameters once they have been estimated. It continues to update them using a gradient descent algorithm such that a geodesic distance measure is minimised. In principle, such models could be parameterised by an initial “momentum” field (Miller et al., 2006, Vaillant et al., 2004), which fully specifies how the velocities – and hence the deformations – evolve over unit time. Unfortunately though, the differential equations involved are difficult to work with, and it is easier to parameterise using a number of velocity fields corresponding to different time periods over the course of the evolution of the diffeomorphism. If u(t) is a velocity field at time t, then the diffeomorphism evolves bydϕdt=u(t)(ϕ(t))Diffeomorphisms are generated by initialising with an identity transform (Φ(0) = x) and integrating over unit time to obtain Φ(1).

The framework described in this paper involves a single flow (velocity) field, which remains constant over unit time. It is similar to the log-Euclidean framework of Arsigny et al., 2006b, Arsigny et al., 2006a. The algorithm is called DARTEL, standing for “Diffeomorphic Anatomical Registration using Exponentiated Lie algebra”.

DARTEL has the advantage, over the small deformation setting, that the resulting deformations are diffeomorphic, easily invertible and can be rapidly computed. It does, however, have a number of disadvantages when compared to variable velocity models. To further understand these limitations, one needs to consider a single point in a brain as the deforming image evolves over unit time. As this point passes different locations of the flow field, then it will be assigned different velocities. Therefore, each of the parameters of such a model will relate to a position in the background space over which the brain deforms, rather than to points within the brain itself. Each voxel in the flow field corresponds to different brain structures at different times during the propagation of the deforming image. Because there is no simple association between a point in the flow field, with a point in the brain, this makes the model parameterisation less ideally suited to computational anatomy studies.

The parameterisation of the variable velocity framework has a more useful physical interpretation, which relates to the velocity of each point in the brain at each time during the course of the evolution. Registration involves simultaneously minimising a measure of difference between the image and the warped template, while also minimising an “energy” measure of the deformations used to warp the template. This energy, often thought of as a squared geodesic distance, is obtained by integrating the energy of the velocity fields over unit time. The fixed velocity field used by DARTEL has to encode the whole trajectory of an evolving diffeomorphism. This constraint may force the diffeomorphism to take very circuitous and high energy trajectories in order to achieve good correspondence between images. In fact, some diffeomorphic configurations, which would easily be achieved if velocities could vary over time, are impossible to reach using DARTEL's constant velocity framework.

A further limitation of the DARTEL model can be seen by registering an image pair and then registering the same image pair, but after first translating one of the images by a few pixels. Providing translations are not explicitly penalised, an ideal registration approach should produce deformation energy measures that are the same in both cases. Unfortunately, this does not happen within the fixed velocity DARTEL framework. Similarly, the shape of the deforming template at particular times during the evolution of the diffeomorphism is not invariant with respect to such an initial translation.

In the Method section, the basic theory behind the constant velocity framework used by DARTEL will be covered. The remainder of this section describes the algorithm that can be used to warp one image to match another. This algorithm involves optimising an objective function that consists of a prior term and a likelihood term. Optimisation is done using a method that uses the first and second derivatives of these terms, with respect to the parameterisation of the deformation. The large number of parameters means that computationally efficient methods are needed for solving the equations, so there is a specific focus on computationally efficient schemes that can handle extremely large, if sparse, matrices. Although the DARTEL model is technically inferior to variable velocity diffeomorphic models, it does have practical advantages in terms of the speed of execution.

The Results and discussion section applies the DARTEL registration scheme to 471 anatomical MR images. The resulting flow fields are used in order to assess the level of internal consistency of the method. The same 471 MR images are also brought into register with a small-deformation model, and the parameterisation of the small-deformation and DARTEL models is compared in terms of how well the information encoded can be used by pattern recognition procedures. A quantitative comparison of fixed velocity DARTEL registration with variable velocity diffeomorphic registration methods will be left for future work.

Section snippets

Method

The DARTEL model assumes a flow field (u) that remains constant over time. With this model, the differential equation describing the evolution of a deformation isdϕdt=u(ϕ(t))Generating a deformation involves starting with an identity transform (Φ(0) = x) and integrating over unit time to obtain Φ(1). The Euler method is a simple integration approach, which involves computing new solutions after many successive small time-steps (h).ϕ(t+h)=ϕ(t)+hu(ϕ(t))Each of these Euler steps is equivalent toϕ(t+h

Results and discussion

Evaluation of warping methods is a complex area. Generally, the results of an evaluation are specific only to the data used to evaluate the model. MR images vary a great deal with different subjects, field strengths, scanners, sequences, etc., so a model that is good for one set of data may not be appropriate for another. Validation should therefore relate to both the data and the algorithm. The question should be about whether it is appropriate to apply a model to a data set, given the

Conclusions

In this paper, we have described DARTEL, a principled and efficient diffeomorphic framework for registering images. Optimisation is performed by a Levenberg–Marquardt strategy, and requires matrix solutions for some very large sparse matrices. The main contribution of this work is the efficient recursive approach used to compute the first and second derivatives used by the optimisation, and the use of a full-multigrid method for solving the equations. This report has focused on underlying

Acknowledgments

I would like to thank Prof. Karl Friston and three reviewers for reading through this manuscript and suggesting a number of improvements. This work was supported by the Wellcome Trust, and much of the writing was done while based in the Psychology Department at Maastricht University.

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